---
lecture: 5
date: 2025-01-20
---
> [!info]
> A normed space is a [[Banach Space]] when the corresponding metric space is complete. Any normed space can be completed to a [[Banach Space]], see the real numbers from $\mathbb{Q}$.

> [!example] Example: $\mathbb{C}^n$ vector space, then
> 1. $\| \, v \, \|_{1} \equiv \Sigma_{k=1}^{n} |v_{k}|, \; v=(v_{1}, \dots, v_{n})$
> 2. $\| \, v \, \|_{\infty} \equiv \text{sub}_{k=1,\dots,n}\{ |v_{k} | \}$
> 
> Which are norms on $\mathbb{C}^n$
## Example
> [!question]
> Consider $C_{c}(X) \subset \Pi_{X}\mathbb{C} = \{ f : X \to C \}$ on functions $X -> \mathbb{C}$ that are non-zero for finitely only many $x \in X$.

Then $C_{c}(X)$ is a vector space under op.

$$f \neq \text{only on} \, Y \, \subset \, X$$
$$q \neq \, \text{only on} \, Z \subset X$$
$$\implies f+q \neq 0 \, \text{only on} \, Y \cup Z$$
$$a \times f \neq \, \text{only on} \, Y$$
with $\delta_{x}(y) = \begin{cases}1, & x=y\\ 0, & x\neq y\end{cases}$
Has linear basis $\{ \delta_{x} \}$, $x \in X$
### Proof
Given that $f \in C_{c}(X)$ then

$f = \Sigma_{x \in X} f(x) \times \delta_{x}$ is a finite sum, and the only possibility.

[[QED]]

So $\text{dim} C_{c} (X) = |X|$.

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Definite norms on $C_{c}(X)$ by $\| \, f \, \|_{1} = \Sigma_{x \in X} | f(x) |$ and $\| \, f \, \|_{\infty} = \text{sup}_{x \in X} | f(x) |$ when $|X| = n$ we recover $\mathbb{C}^n$.

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We write $V \simeq W$and say that $V$ and $W$ are [[Isomorphic]]
1. **As sets** if $\exists$ [[Bijective|bijection]] $f : V \to W$
2. **As vector spaces** if in addition $f$ is **linear**; $f(a \times u + b \times v) = a \times f(u) + b \times f(v), \; \forall a,b \in \mathbb{C}, \; u,v \in V$
3. **As [[Normed vector Space|normed spaces]]** if in addition $f$ is **[[Isometric]]**; $\| \, f(v) \, \| \, = \| \, v \, \|, \; \forall v \in V$.
$f$ should preserve all relevant structure.
# [[Hilbert Spaces]]
# Triangle Inequality
Follows from the [[Cauchy-Schwarz Inequality]]

> [!example] Example on $C_{c}(X)$ then
> $$(f | g) = \Sigma_{x \in X} f(x) \times \overline{g(x)}$$
> This defines an [[Inner Product|inner product]], which can be completed to a [[Hilbert Spaces|Hilbert space]].
> $$\| \, f \, \|_{2} = (\Sigma_{x \in X} | f(x) |^2)^{\frac{1}{2}}$$
> > [!note] The [[Inner Product|inner product]] here
> > $$\mathbb{C}^{n} (u | v) = \Sigma_{k = 1}^{n} u_{k} \overline{v_{k}}$$
> > $$\mathbb{R}^{n} \| \, u \, \|_{2} = (\Sigma |u_{k}|^{2})^{\frac{1}{2}}$$
> > $$\vec{u} \times \vec{v} = (\vec{u} | \vec{v})$$